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This paper presents a 3D mesh adaptivity strategy on unstructured tetrahedralmeshes by a posteriori error estimates based on metrics, studied on the case ofa nonlinear finite element minimization scheme for the Landau-de Gennes freeenergy functional of nematic liquid crystals. Newton's iteration for tensorfields is employed with steepest descent method possibly stepping in.  Aspects relating the driving of mesh adaptivity within the nonlinear schemeare considered. The algorithmic performance is found to depend on at least twofactors: when to trigger each single mesh adaptation, and the precision of thecorrelated remeshing. Each factor is represented by a parameter, with itsvalues possibly varying for every new mesh adaptation. We empirically show thatthe time of the overall algorithm convergence can vary considerably whendifferent sequences of parameters are used, thus posing a question aboutoptimality.  The extensive testings and debugging done within this work on the simulationof systems of nematic colloids substantially contributed to the upgrade of anopen source finite element-oriented programming language to its 3D meshingpossibilities, as also to an outer 3D remeshing module.

A mesh adaptivity scheme on the Landau-de Gennes functional minimization case in 3D, and its driving efficiency